<!-- build time:Wed Jun 21 2023 22:33:34 GMT+0800 (GMT+08:00) --><!DOCTYPE html><html lang="zh-CN"><head><meta charset="UTF-8"><meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=2"><meta name="theme-color" content="#FFF"><meta name="baidu-site-verification" content="code-C0oocRvMWv"><link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon.png"><link rel="icon" type="image/ico" sizes="32x32" href="/images/favicon.ico"><link rel="mask-icon" href="/images/logo.svg" color=""><link rel="manifest" href="/images/manifest.json"><meta name="msapplication-config" content="/images/browserconfig.xml"><meta http-equiv="Cache-Control" content="no-transform"><meta http-equiv="Cache-Control" content="no-siteapp"><meta name="baidu-site-verification" content="https://jiang-hs.gitee.io"><link rel="alternate" type="application/rss+xml" title="航 順" href="https://jiang-hs.gitee.io/rss.xml"><link rel="alternate" type="application/atom+xml" title="航 順" 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property="og:url" content="https://jiang-hs.gitee.io/posts/ae4d52e6/index.html"><meta property="og:site_name" content="航 順"><meta property="og:description" content="# 1 Abstract 我们介绍了强化学习在间接机制中的应用，与现有的序列价格机制一起工作，它概括了序列独裁和贴出价格机制，并且本质上刻画了所有强有力的显而易见的策略证明机制。在这个类中学习一个最优机制形成了一个部分可观察的马尔可夫决策过程。我们为这类机制何时比简单的静态机制更强大、学习的观察统计的充分性或不足以及复杂 (深度) 策略的必要性提供了严格的条件。我们表明，我们的方法可以在几个实验环"><meta property="og:locale" content="zh_CN"><meta property="og:image" content="https://shun309.oss-cn-hangzhou.aliyuncs.com/photos/20210607214021.png"><meta property="article:published_time" content="2021-06-07T12:24:06.000Z"><meta property="article:modified_time" content="2023-01-12T04:19:43.105Z"><meta property="article:author" content="hang shun"><meta property="article:tag" content="hang shun"><meta name="twitter:card" content="summary"><meta name="twitter:image" content="https://shun309.oss-cn-hangzhou.aliyuncs.com/photos/20210607214021.png"><title>顺序价格机制的强化学习 | hang shun = 航 順 = 天官赐福，百无禁忌</title><meta name="generator" content="Hexo 5.4.2"></head><body itemscope itemtype="http://schema.org/WebPage"><div id="loading"><div class="cat"><div class="body"></div><div class="head"><div class="face"></div></div><div class="foot"><div class="tummy-end"></div><div class="bottom"></div><div class="legs left"></div><div class="legs right"></div></div><div class="paw"><div class="hands left"></div><div class="hands right"></div></div></div></div><div id="container"><header id="header" itemscope itemtype="http://schema.org/WPHeader"><div class="inner"><div id="brand"><div class="pjax"><h1 itemprop="name headline">顺序价格机制的强化学习</h1><div class="meta"><span class="item" title="创建时间：2021-06-07 20:24:06"><span class="icon"><i class="ic i-calendar"></i> </span><span class="text">发表于</span> <time itemprop="dateCreated datePublished" datetime="2021-06-07T20:24:06+08:00">2021-06-07</time> </span><span class="item" title="本文字数"><span class="icon"><i class="ic i-pen"></i> </span><span class="text">本文字数</span> <span>6.2k</span> <span class="text">字</span> </span><span class="item" title="阅读时长"><span class="icon"><i class="ic i-clock"></i> </span><span class="text">阅读时长</span> <span>6 分钟</span></span></div></div></div><nav id="nav"><div class="inner"><div class="toggle"><div class="lines" aria-label="切换导航栏"><span class="line"></span> <span class="line"></span> <span class="line"></span></div></div><ul class="menu"><li class="item title"><a href="/" rel="start">hang shun</a></li></ul><ul class="right"><li class="item theme"><i class="ic i-sun"></i></li><li class="item search"><i class="ic i-search"></i></li></ul></div></nav></div><div id="imgs" class="pjax"><ul><li class="item" data-background-image="https://pic1.imgdb.cn/item/60d7f9475132923bf8a8a276.jpg"></li><li class="item" data-background-image="https://pic1.imgdb.cn/item/60d7f93d5132923bf8a86d6c.jpg"></li><li class="item" data-background-image="https://pic1.imgdb.cn/item/60d7f94f5132923bf8a8ce66.jpg"></li><li class="item" data-background-image="https://pic1.imgdb.cn/item/60d7f94f5132923bf8a8cd0f.jpg"></li><li class="item" data-background-image="https://pic1.imgdb.cn/item/64427c380d2dde5777afaa9c.jpg"></li><li class="item" data-background-image="https://pic1.imgdb.cn/item/64427e500d2dde5777b2ca50.jpg"></li></ul></div></header><div id="waves"><svg class="waves" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 24 150 28" preserveAspectRatio="none" shape-rendering="auto"><defs><path id="gentle-wave" d="M-160 44c30 0 58-18 88-18s 58 18 88 18 58-18 88-18 58 18 88 18 v44h-352z"/></defs><g class="parallax"><use xlink:href="#gentle-wave" x="48" y="0"/><use xlink:href="#gentle-wave" x="48" y="3"/><use xlink:href="#gentle-wave" x="48" y="5"/><use xlink:href="#gentle-wave" x="48" y="7"/></g></svg></div><main><div class="inner"><div id="main" class="pjax"><div class="article wrap"><div class="breadcrumb" itemscope itemtype="https://schema.org/BreadcrumbList"><i class="ic i-home"></i> <span><a href="/">首页</a></span></div><article itemscope itemtype="http://schema.org/Article" class="post block" lang="zh-CN"><link itemprop="mainEntityOfPage" href="https://jiang-hs.gitee.io/posts/ae4d52e6/"><span hidden itemprop="author" itemscope itemtype="http://schema.org/Person"><meta itemprop="image" content="/images/avatar.jpg"><meta itemprop="name" content="hang shun"><meta itemprop="description" content="天官赐福，百无禁忌, 世中逢尔，雨中逢花"></span><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="航 順"></span><div class="body md" itemprop="articleBody"><h1 id="1-abstract"><a class="anchor" href="#1-abstract">#</a> 1 Abstract</h1><p>我们介绍了强化学习在间接机制中的应用，与现有的序列价格机制一起工作，它概括了序列独裁和贴出价格机制，并且本质上刻画了所有强有力的显而易见的策略证明机制。在这个类中学习一个最优机制形成了一个部分可观察的马尔可夫决策过程。我们为这类机制何时比简单的静态机制更强大、学习的观察统计的充分性或不足以及复杂 (深度) 策略的必要性提供了严格的条件。我们表明，我们的方法可以在几个实验环境中学习最佳或接近最佳的机制。</p><h1 id="2-preliminaries"><a class="anchor" href="#2-preliminaries">#</a> 2 Preliminaries</h1><p><ins><strong>经济框架</strong></ins> ：有<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">n</span></span></span></span> 个代理和<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">m</span></span></span></span> 个不可分割的项目。设<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[n] = \{1,···,n\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">[</span><span class="mord mathnormal">n</span><span class="mclose">]</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">{</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">n</span><span class="mclose">}</span></span></span></span><strong> 是代理的集合</strong>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[m]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mclose">]</span></span></span></span><strong> 是项目的集合</strong>。代理有一个<strong>估价函数</strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub><mo>:</mo><msup><mn>2</mn><mrow><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><msub><mi>R</mi><mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">V_i:2^{[m]}→R_{≥0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.22222em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">:</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.8879999999999999em;vertical-align:0"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8879999999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">[</span><span class="mord mathnormal mtight">m</span><span class="mclose mtight">]</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.928509em;vertical-align:-.24517899999999998em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.00773em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mrel mtight">≥</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.24517899999999998em"><span></span></span></span></span></span></span></span></span></span>，它将一捆捆物品映射为真实价值。特殊的，单位需求估价是这样一种估价，在这种估价中，代理对每一个项目都有一个值，而一个包的值就是该包中一个项目的最大值。设<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">v</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>v</mi><mn>1</mn></msub><mo separator="true">,</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">,</mo><msub><mi>v</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{v} = (v_1,···,v_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.151392em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 表示<strong>估价概况</strong>。我们假设<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">v</mi></mrow><annotation encoding="application/x-tex">\mathrm{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span></span></span></span> 是从一个可能相关的值分布<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.02778em">D</span></span></span></span> 中采样的。设计者可以通过来自联合分布的样本来访问这个分布<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.02778em">D</span></span></span></span>。</p><p><strong>分配</strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">x</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{x} = (x_1,···,x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord"><span class="mord mathrm">x</span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.151392em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是不相交的项目束的轮廓<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>∩</mo><msub><mi>x</mi><mi>j</mi></msub><mo>=</mo><mtext>∅</mtext></mrow><annotation encoding="application/x-tex">(x_i∩ x_j=∅</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.716668em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.80556em;vertical-align:-.05556em"></span><span class="mord">∅</span></span></span></span> 对于每个<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo mathvariant="normal">≠</mo><mi>j</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i≠j∈[n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">[</span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span>)，其中<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>⊆</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x_i⊆ [m]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7859700000000001em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mclose">]</span></span></span></span> 是分配给代理<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 的项目集</p><p>一种经济机制与代理人相互作用并决定一种结果，即分配<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">x</mi></mrow><annotation encoding="application/x-tex">\mathrm{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord"><span class="mord mathrm">x</span></span></span></span></span> 和<strong>转移 (支付)</strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>τ</mi><mn>1</mn></msub><mo separator="true">,</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">,</mo><msub><mi>τ</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{τ} = (τ_1,···,τ_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-.1132em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.151392em"><span style="top:-2.5500000000000003em;margin-left:-.1132em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，其中<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>τ</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">τ_i≥ 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7859700000000001em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.1132em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">0</span></span></span></span> 是代理人<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 的支付。我们通过<strong>目标函数</strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo separator="true">,</mo><mi>τ</mi><mo separator="true">;</mo><mi mathvariant="normal">v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\mathrm{x},\mathrm{τ};\mathrm{v})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">x</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span><span class="mclose">)</span></span></span></span> 来测量在估值曲线<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03588em">v</span></span></span></span> 下机制结果<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">x</mi><mo separator="true">,</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">\mathrm{x},\mathrm{τ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.625em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathrm">x</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span></span></span></span></span> 的性能。</p><p><ins><strong>目标</strong></ins>：设计一种机制，其结果使期望值最大化，这是关于价值分布的目标函数。</p><p>我们的框架允许不同的目标，例如:</p><ul><li>社会福利：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo separator="true">,</mo><mi>τ</mi><mo separator="true">;</mo><mi mathvariant="normal">v</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msub><msub><mi>v</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\mathrm{x},\mathrm{τ};\mathrm{v})= \sum_{i∈[n]}v_i(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">x</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.22471em;vertical-align:-.47471em"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.22528999999999993em"><span style="top:-2.40029em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mopen mtight">[</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.47471em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，</li><li>税收：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo separator="true">,</mo><mi>τ</mi><mo separator="true">;</mo><mi mathvariant="normal">v</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msub><msub><mi>τ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">g(\mathrm{x},\mathrm{τ};\mathrm{v})=\sum_{i∈[n]}τ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">x</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.22471em;vertical-align:-.47471em"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.22528999999999993em"><span style="top:-2.40029em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mopen mtight">[</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.47471em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.1132em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span>，</li><li>最大 - 最小公平性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo separator="true">,</mo><mi>τ</mi><mo separator="true">;</mo><mi mathvariant="normal">v</mi><mo stretchy="false">)</mo><mi>m</mi><mi>i</mi><msub><mi>n</mi><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msub><msub><mi>v</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\mathrm{x},\mathrm{τ};\mathrm{v})min_{i∈[n]}v_i(x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-.3551999999999999em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">x</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span><span class="mclose">)</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.34480000000000005em"><span style="top:-2.5198em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">∈</span><span class="mopen mtight">[</span><span class="mord mathnormal mtight">n</span><span class="mclose mtight">]</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.3551999999999999em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-.03588em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。</li></ul><p><ins><strong>Sequential Price Mechanisms (顺序价格机制)</strong></ins>：我们研究<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>P</mi><mi>M</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">SPMs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal">s</span></span></span></span> 家族。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>P</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">SPM</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.10903em">M</span></span></span></span> 跨回合与代理交互，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>∈</mo><mrow><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo></mrow></mrow><annotation encoding="application/x-tex">t ∈ {1,2,···}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65418em;vertical-align:-.0391em"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.8388800000000001em;vertical-align:-.19444em"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span></span></span></span></span>，并在每一轮访问不同的代理。在第<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.61508em;vertical-align:0"></span><span class="mord mathnormal">t</span></span></span></span> 轮结束时，该机制保持以下参数：访问的第一批<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.61508em;vertical-align:0"></span><span class="mord mathnormal">t</span></span></span></span> 个代理的<strong>临时分配</strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">x^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span>、<strong>临时支付</strong>配置文件<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>τ</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">τ^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span> 和剩余设置<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ρ</mi><mi>t</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>ρ</mi><mrow><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mrow><mi>t</mi></msubsup><mtext>，</mtext><msubsup><mi>ρ</mi><mrow><mi>i</mi><mi>t</mi><mi>e</mi><mi>m</mi><mi>s</mi></mrow><mi>t</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ρ^t= (ρ^t_{agents}，ρ^t_{items})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.9879959999999999em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.176664em;vertical-align:-.383108em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-2.4530000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight" style="margin-right:.03588em">g</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.383108em"><span></span></span></span></span></span></span><span class="mord cjk_fallback">，</span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-2.441336em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">m</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.258664em"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，其中<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>ρ</mi><mrow><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mrow><mi>t</mi></msubsup><mo>⊆</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">ρ^t_{agents}⊆ [n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.176664em;vertical-align:-.383108em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-2.4530000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight" style="margin-right:.03588em">g</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">[</span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span><strong> 和<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>ρ</mi><mrow><mi>i</mi><mi>t</mi><mi>e</mi><mi>m</mi><mi>s</mi></mrow><mi>t</mi></msubsup><mo>⊆</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">ρ^t_{items}⊆ [m]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.05222em;vertical-align:-.258664em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-2.441336em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">m</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mclose">]</span></span></span></span> 分别是尚未访问的代理集和仍然可用的项目</strong>。在每一轮<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.61508em;vertical-align:0"></span><span class="mord mathnormal">t</span></span></span></span> 中，</p><ul><li>(1) 该机制选择一个代理<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>i</mi><mi>t</mi></msup><mo>∈</mo><msubsup><mi>ρ</mi><mrow><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">i^t∈ρ^{t-1}_{agents}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.832656em;vertical-align:-.0391em"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.236103em;vertical-align:-.381864em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.854239em"><span style="top:-2.454244em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight" style="margin-right:.03588em">g</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.1031310000000003em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.381864em"><span></span></span></span></span></span></span></span></span></span>，为每个可用项目<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>∈</mo><msubsup><mi>ρ</mi><mrow><mi>i</mi><mi>t</mi><mi>e</mi><mi>m</mi><mi>s</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">j∈ρ^{t-1}_{items}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathnormal" style="margin-right:.05724em">j</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.131103em;vertical-align:-.276864em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.854239em"><span style="top:-2.4231360000000004em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">m</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.1031310000000003em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.276864em"><span></span></span></span></span></span></span></span></span></span> 项目发布价格 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>p</mi><mi>j</mi><mi>t</mi></msubsup></mrow><annotation encoding="application/x-tex">p^t_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1883279999999998em;vertical-align:-.394772em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-2.441336em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.394772em"><span></span></span></span></span></span></span></span></span></span>;</li><li>(2) 代理<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>i</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">i^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span> 从可用项目集合中选择一个束<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">x^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span>，并收取费用<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mo>∑</mo><mrow><mi>j</mi><mo>∈</mo><msup><mi>x</mi><mi>t</mi></msup></mrow></msub><msubsup><mi>p</mi><mi>j</mi><mi>t</mi></msubsup></mrow><annotation encoding="application/x-tex">\sum_{j∈x^t}p^t_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.229374em;vertical-align:-.43581800000000004em"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.20802999999999994em"><span style="top:-2.40029em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7253428571428571em"><span style="top:-2.786em;margin-right:.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.43581800000000004em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-2.441336em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:.05724em">j</span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.394772em"><span></span></span></span></span></span></span></span></span></span>;</li><li>(3) 剩余项目、剩余代理、临时分配和临时支付配置文件都会相应地更新。</li></ul><p>这里，可以方便地初始化：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>ρ</mi><mrow><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mrow><mn>0</mn></msubsup><mo>=</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo separator="true">,</mo><msubsup><mi>ρ</mi><mrow><mi>i</mi><mi>t</mi><mi>e</mi><mi>m</mi><mi>s</mi></mrow><mn>0</mn></msubsup><mo>=</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mo separator="true">,</mo><msup><mi mathvariant="normal">x</mi><mi>t</mi></msup><mo>=</mo><mo stretchy="false">(</mo><mtext>∅</mtext><mo separator="true">,</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">,</mo><mtext>∅</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ρ^0_{agents}= [n], ρ^0_{items}= [m],\mathrm{x}^t= (∅,···,∅)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.197216em;vertical-align:-.383108em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-2.4530000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="mord mathnormal mtight" style="margin-right:.03588em">g</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.072772em;vertical-align:-.258664em"></span><span class="mopen">[</span><span class="mord mathnormal">n</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-2.441336em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight">t</span><span class="mord mathnormal mtight">e</span><span class="mord mathnormal mtight">m</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.043556em;vertical-align:-.25em"></span><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord"><span class="mord mathrm">x</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord">∅</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">∅</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>τ</mi><mn>0</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">⋅</mo><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">τ^0= (0,··· ,0).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8141079999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">0</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></p><p><ins><strong>Learning Framework (学习框架)</strong></ins>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>P</mi><mi>M</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">SPMs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal">s</span></span></span></span> 的顺序性，以及代理人估价的私有性，使得将自动机制设计问题公式化为部分可观察的马尔可夫决策过程 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>O</mi><mi>M</mi><mi>D</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">POMDP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal" style="margin-right:.02778em">D</span><span class="mord mathnormal" style="margin-right:.13889em">P</span></span></span></span>) 是有用的。<strong>一个<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>O</mi><mi>M</mi><mi>D</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">POMDP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal" style="margin-right:.02778em">D</span><span class="mord mathnormal" style="margin-right:.13889em">P</span></span></span></span> 是一个<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>D</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">MDP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal" style="margin-right:.02778em">D</span><span class="mord mathnormal" style="margin-right:.13889em">P</span></span></span></span> (由一个状态空间<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span></span></span></span>，一个动作空间<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal">A</span></span></span></span>，一个马尔科夫状态 - 动作 - 状态转移概率函数<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">P</mi><mo stretchy="false">(</mo><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">;</mo><mi>s</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}(s&#x27;;s,a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-.25em"></span><span class="mord"><span class="mord mathbb">P</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.751892em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> 和奖励函数<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>s</mi><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(s,a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">r</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> 构成)，以及从每个动作和结果状态到<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">P</mi><mo stretchy="false">(</mo><mi>o</mi><mo separator="true">;</mo><msup><mi>s</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}(o;s&#x27;,a)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-.25em"></span><span class="mord"><span class="mord mathbb">P</span></span><span class="mopen">(</span><span class="mord mathnormal">o</span><span class="mpunct">;</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.751892em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">a</span><span class="mclose">)</span></span></span></span> 给出的观测值<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi></mrow><annotation encoding="application/x-tex">o</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">o</span></span></span></span> 的可能随机映射</strong>。</p><p>对于<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>P</mi><mi>M</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">SPMs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal">s</span></span></span></span>，<strong>状态</strong>对应于仍未分配的项目、尚未访问的代理、部分分配和代理的评估功能。一个<strong>动作</strong>决定了下一步要去哪个代理商以及设定什么价格。这导致新的状态和观察，即代理挑选的项目。通过这种方式，状态转移由代理策略控制，也就是说，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>P</mi><mi>M</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">SPMs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal">s</span></span></span></span> 的主导策略均衡。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>o</mi><mi>l</mi><mi>i</mi><mi>c</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">policy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">p</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span><span class="mord mathnormal" style="margin-right:.03588em">y</span></span></span></span> 定义了<strong>机制</strong>的规则。适当定义的奖励函数的最优<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>o</mi><mi>l</mi><mi>i</mi><mi>c</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">policy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">p</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span><span class="mord mathnormal" style="margin-right:.03588em">y</span></span></span></span> 对应于最优<strong>机制</strong>。求解<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>O</mi><mi>M</mi><mi>D</mi><mi>P</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">POMDPs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal" style="margin-right:.02778em">D</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal">s</span></span></span></span> 需要对<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mi>e</mi><mi>l</mi><mi>i</mi><mi>e</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">belief</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">i</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:.10764em">f</span></span></span></span> 状态进行推理，即在给定观测历史的<strong>状态</strong>分布的<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mi>e</mi><mi>l</mi><mi>i</mi><mi>e</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">belief</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">i</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:.10764em">f</span></span></span></span>。一个典型的方法是为<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mi>e</mi><mi>l</mi><mi>i</mi><mi>e</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">belief</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">b</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">i</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:.10764em">f</span></span></span></span> 状态找到一个充分的统计量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi>o</mi><mi>l</mi><mi>i</mi><mi>c</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">policy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8888799999999999em;vertical-align:-.19444em"></span><span class="mord mathnormal">p</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">i</span><span class="mord mathnormal">c</span><span class="mord mathnormal" style="margin-right:.03588em">y</span></span></span></span> 被定义为从这个统计量到动作的映射。</p><h1 id="3-characterization-results"><a class="anchor" href="#3-characterization-results">#</a> 3 Characterization Results</h1><p>表征结果</p><p>在<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>P</mi><mi>M</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">SPMs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal">s</span></span></span></span> 中，前几轮的结果可用于决定拜访哪个代理商以及在当前一轮中设定什么价格。这使得价格可以个性化和自适应，也使得访问代理的顺序可以自适应。我们接下来介绍一些特殊情况。</p><p><strong>定义 1</strong> (匿名静态价格 (ASP) 机制)。价格是在开始时设定的 (以一种潜在的随机方式)，并且在各轮之间以及对每个代理都是相同的。</p><p><strong>定义 2</strong> (个性化静态价格 (PSP) 机制)。价格是在开始时设定的 (以一种潜在的随机方式)，并且在各轮之间是相同的，但是每个代理可能面临不同的价格。除了价格，我们还对机制选择代理的顺序感兴趣:</p><p><strong>定义 3</strong> (静态顺序 (so) 机制)。顺序是在开始时设定的 (以一种潜在的随机方式)，不会在各轮之间改变。</p><p>我们在图 1 中说明了各种机制类之间的关系。</p><p>ASP 类是 PSP 类的子集，PSP 类是 SPM 的子集，1 序列独裁 (SD) 机制是 ASP 的子集 (所有支付都设置为零)，可能有自适应或静态顺序。随机序列独裁机制 (RSD) 位于 SD 和静态秩序 (SO) 的交叉点。</p><p><img data-src="https://shun309.oss-cn-hangzhou.aliyuncs.com/photos/20210607214021.png" alt=""></p><p>图 1. 顺序价格机制的分类</p><h2 id="31-个性化价格和适应性的需求"><a class="anchor" href="#31-个性化价格和适应性的需求">#</a> 3.1 个性化价格和适应性的需求</h2><p>在这一节中，我们展示了个性化价格和适应性对于优化福利是必要的，即使是在令人惊讶的简单设置中。这进一步激发了将设计问题公式化为<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>O</mi><mi>M</mi><mi>D</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">POMDP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal" style="margin-right:.02778em">D</span><span class="mord mathnormal" style="margin-right:.13889em">P</span></span></span></span> 并使用<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">RL</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.00773em">R</span><span class="mord mathnormal">L</span></span></span></span> 方法来解决它的动机。我们在实验工作中回到这些命题的证明中所包含的例子。</p><p>定义一个福利最优的特殊目的群体模型，使其成为一种机制，在该类<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>M</mi><mi>P</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">SMPs</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal">s</span></span></span></span> 中优化预期的社会福利。</p><p><strong>命题一：存在一个带有一个项目和两个 IID 代理的设置，其中福利最优的<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>P</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">SPM</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.10903em">M</span></span></span></span> 机制必须使用个性化的价格。</strong></p><p>证明：考虑一个包含一个项目和两个 IID 代理的设置，其中每个代理都有一个均匀分布在该设置上的估价 {1，3}。请注意，WLOG 只考虑 0 和 2 的价格。一个最优机制首先以价格 p1= 2 向代理 1 提供物品。然后，如果该物品仍然可用，则该机制以价格 p2= 0 将该物品提供给代理 2。没有一个价格 p 能达到 OPT。如果 p = 0，第一个被访问的代理可能会在他们的值为 1 而另一个代理的值为 3 时获取该项目。如果 p = 2，则如果两个代理的值都为 1，则该项目将未分配。</p><p>请注意，在命题 1 的证明中使用的示例中，自适应订单不会消除个性化价格的需要。有趣的是，即使有 IID 代理商和相同的商品，我们也需要价格合适的特殊产品。</p><p><strong>命题二：存在两个相同项目和三个 IID 代理的单位需求设置，其中福利最优的特殊产品模型必须使用自适应价格。</strong></p><p>我们提供了一个证据草图，并将证据推迟到附录 A。对适应性价格的需求来自于在第一个代理人做出决定后对剩余物品的供应做出响应的需求：(1) 如果这个代理人购买，那么在剩下一个物品和两个代理人的情况下，最优价格应该足够高，以将该物品分配给高价值的代理人，或者 (2) 如果这个代理人不购买，后续价格应该较低，以确保剩余的两个物品都被分配。</p><p>以下命题表明，即使最优价格是匿名的和静态的，自适应订单也是必要的。</p><p><strong>命题三：存在一个单位需求设置，有两个相同的项目和六个具有相关估值的代理，其中福利最优的特殊产品模型必须使用自适应订单 (但匿名静态价格就足够了)。</strong></p><p>我们将证明推迟到附录 A。直觉是代理人的估价是依赖的，知道一个特定代理人的价值可以对其他代理人的价值的条件分布提供重要的见解。这个 “bellweather” 代理的价值可以从他们是否购买的决定中推断出来，这个额外的推断对于最优地订购剩余的代理是必要的。因此，机制的顺序必须适应该代理的决定。</p><p>即使当项目是相同的，并且代理人的价值分布是独立的，适应性订单和适应性价格都可能是必要的。</p><p><strong>命题四：存在一个单位需求设置，其中两个相同的项目和四个代理具有独立 (非相同) 分布的价值，其中福利最优的特殊产品模型必须使用自适应订单和自适应价格。</strong></p><p>我们将证据推迟到附录 A。直觉是，与其他一些代理相比，一个代理的价值 “上限” 和 “下限” 都更高。该机制最好访问其他代理，以确定向该特定代理提供的最佳价格，这种信息收集过程可能需要很长时间。我们在附录 D 中给出了 SPMs 的额外的、细颗粒的结果，说明了 SPMs 试剂的快速或适应性订购</p><h1 id="4-learning-optimal-spms"><a class="anchor" href="#4-learning-optimal-spms">#</a> 4 Learning Optimal SPMs</h1><p>在这一节中，我们将设计一个最优的 SPM 问题转化为一个<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>O</mi><mi>M</mi><mi>D</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">POMDP</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.13889em">P</span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mord mathnormal" style="margin-right:.10903em">M</span><span class="mord mathnormal" style="margin-right:.02778em">D</span><span class="mord mathnormal" style="margin-right:.13889em">P</span></span></span></span> 问题。我们的讨论主要涉及福利最大化，但我们也将评论我们的结果如何扩展到收入最大化和最大 - 最小公平。</p><p>我们对 POMDP 的定义如下:</p><ul><li>状态<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>s</mi><mi>t</mi></msup><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="normal">v</mi><mo separator="true">,</mo><msup><mi mathvariant="normal">x</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo separator="true">,</mo><msup><mi>ρ</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s^t= (\mathrm{v},\mathrm{x}^{t-1},ρ^{t-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord"><span class="mord mathrm">x</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是一个元组，由代理评估<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">v</mi></mrow><annotation encoding="application/x-tex">\mathrm{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span></span></span></span>、当前部分分配<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">x</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathrm{x}^{t-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8141079999999999em;vertical-align:0"></span><span class="mord"><span class="mord"><span class="mord mathrm">x</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 和剩余设置<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ρ</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">ρ^{t-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 组成，剩余设置<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ρ</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">ρ^{t-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 由尚未访问的代理和尚未分配的项目组成。</li><li>在<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>a</mi><mi>t</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>i</mi><mi>t</mi></msup><mo separator="true">,</mo><msup><mi>p</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a^t= (i^t,p^t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.043556em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 定义了下一个选定的代理<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>i</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">i^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span> 和发布的价格<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>p</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">p^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.9879959999999999em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span>。</li><li>对于状态转换，所选代理选择一个项目或一捆项目<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">x^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span>，产生新的状态<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>s</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">s^{t+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8141079999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>，其中一捆项目<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">x^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span> 被添加到部分分配<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">x</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathrm{x}^{t- 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8141079999999999em;vertical-align:0"></span><span class="mord"><span class="mord"><span class="mord mathrm">x</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 以形成新的部分分配<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">x</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">\mathrm{x}^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord"><span class="mord mathrm">x</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span>，并且项和代理从剩余设置<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ρ</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">ρ^{t-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 中移除，形成<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ρ</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">ρ^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.9879959999999999em;vertical-align:-.19444em"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span>。</li><li>观察<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>o</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>x</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">o^{t+1} = x^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8141079999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">o</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span> 由在第<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.61508em;vertical-align:0"></span><span class="mord mathnormal">t</span></span></span></span> 轮选择的代理选择的一个或一组项目<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>t</mi></msup></mrow><annotation encoding="application/x-tex">x^t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7935559999999999em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.7935559999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span> 组成。</li><li>当机制结果<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">x</mi></mrow><annotation encoding="application/x-tex">\mathrm{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord"><span class="mord mathrm">x</span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\mathrm{τ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span></span></span></span></span> 有效时，我们只在终端状态下提供奖励。这些终端奖励给<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="normal">x</mi><mo separator="true">,</mo><mi>τ</mi><mo separator="true">;</mo><mi mathvariant="normal">v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\mathrm{x},\mathrm{τ};\mathrm{v})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathrm">x</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.1132em">τ</span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathrm" style="margin-right:.01389em">v</span></span><span class="mclose">)</span></span></span></span> 也就是我们想要最大化的目标函数</li></ul><p>接下来，我们研究在任何观察历史之后足以确定最佳行动的信息。我们表明，对于单位需求估值和社会福利目标的情况，分析基本上是严密的。我们把证据推迟到附录二</p><p><strong>命题五：对于具有独立 (非相同) 分布估值的代理，以福利或收入最大化为目标，维持剩余代理和项目足以确定最优策略。</strong></p><p>有趣的是，命题 5 中的陈述在处理更敏感的分配目标 (如最大 - 最小公平) 时不再成立。3 下一个定理说明了所有分配和目标的历史信息。</p><p><strong>定理 1</strong>：在相关估价的情况下，无论设计目标是什么，分配矩阵以及尚未收到报价的代理都足以确定最优策略。此外，存在具有相关估值的单位需求设置，其中最优策略必须使用大小为<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Ω</mtext><mo stretchy="false">(</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">{</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">}</mo><mi>l</mi><mi>o</mi><mi>g</mi><mo stretchy="false">(</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">{</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ω (min\{n, m\}log (max\{n, m\}))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">Ω</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">n</span><span class="mopen">{</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mclose">}</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mopen">{</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mclose">}</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span> 的信息。</p><p>就充分性而言，分配矩阵和剩余代理始终足以恢复任何 (确定性) 策略的整个观察历史。结果如下，因为在给定整个观测历史的情况下，总是存在确定性的最优策略。定理 1 还确定，从空间复杂度的观点来看，携带当前分配和剩余代理是必要的，因为当前分配和剩余代理可以在<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">{</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">}</mo><mi>l</mi><mi>o</mi><mi>g</mi><mo stretchy="false">(</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">{</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(min\{n, m\}log (max\{n, m\}))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mord mathnormal">i</span><span class="mord mathnormal">n</span><span class="mopen">{</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mclose">}</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mord mathnormal">a</span><span class="mord mathnormal">x</span><span class="mopen">{</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathnormal">m</span><span class="mclose">}</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span> 空间中编码。另一个直接推论是，关于剩余代理和项目 (线性空间) 的知识，而不是先前代理的决策，通常不足以支持最优策略。相关估值产生的问题来自于需要推断剩余代理的估值。</p><p>如下一个命题所示，只能访问剩余代理的策略对应于 SPM 的特殊情况。</p><p><strong>命题 6：具有静态 (可能是个性化的) 价格和静态订单的 SPM 子类对应于只能访问剩余代理集的策略。</strong></p><p><strong>线性策略是不够的</strong>。给定分配矩阵和剩余代理的访问权，理解支持福利最优机制所必需的政策类别也是有趣的。给定输入参数<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">x</span></span></span></span>，线性策略使用线性变换<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">⋅</mo><msubsup><mi>θ</mi><mi mathvariant="normal">ℓ</mi><mi>T</mi></msubsup></mrow><annotation encoding="application/x-tex">x ·θ^T_{\ell}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1244389999999997em;vertical-align:-.2831079999999999em"></span><span class="mord mathnormal">x</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.8413309999999999em"><span style="top:-2.4168920000000003em;margin-left:-.02778em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span></span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:.13889em">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.2831079999999999em"><span></span></span></span></span></span></span></span></span></span> 将输入映射到第<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord">ℓ</span></span></span></span> 个输出，其中<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>θ</mi><mi mathvariant="normal">ℓ</mi></msub><msub><mo stretchy="false">}</mo><mi mathvariant="normal">ℓ</mi></msub></mrow><annotation encoding="application/x-tex">θ = \{θ_{\ell}\}_{\ell}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.02778em">θ</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-.02778em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">ℓ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> 是策略的参数。就我们的学习框架而言，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">x</span></span></span></span> 是一个扁平的二进制分配矩阵和剩余代理的二进制向量。我们输出<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n + m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.66666em;vertical-align:-.08333em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathnormal">m</span></span></span></span> 个输出变量，代表代理商的分数 (意味着一个订单) 和商品的价格。我们能够证明线性政策是不够的。</p><p><strong>命题 7：存在这样一种设置，即福利最优的特殊产品模型不能通过在分配矩阵和剩余代理中是线性的策略来实现。</strong></p><p>这为 SPM 设计问题的非线性方法提供了支持，促进了神经网络的使用。</p></div><footer><div class="meta"><span class="item"><span class="icon"><i class="ic i-calendar-check"></i> </span><span class="text">更新于</span> <time title="修改时间：2023-01-12 12:19:43" itemprop="dateModified" datetime="2023-01-12T12:19:43+08:00">2023-01-12</time> </span><span id="posts/ae4d52e6/" class="item leancloud_visitors" data-flag-title="顺序价格机制的强化学习" title="阅读次数"><span class="icon"><i class="ic i-eye"></i> </span><span class="text">阅读次数</span> <span 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Abstract</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#2-preliminaries"><span class="toc-number">2.</span> <span class="toc-text">2 Preliminaries</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#3-characterization-results"><span class="toc-number">3.</span> <span class="toc-text">3 Characterization Results</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#31-%E4%B8%AA%E6%80%A7%E5%8C%96%E4%BB%B7%E6%A0%BC%E5%92%8C%E9%80%82%E5%BA%94%E6%80%A7%E7%9A%84%E9%9C%80%E6%B1%82"><span class="toc-number">3.1.</span> <span class="toc-text">3.1 个性化价格和适应性的需求</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#4-learning-optimal-spms"><span class="toc-number">4.</span> <span class="toc-text">4 Learning Optimal SPMs</span></a></li></ol></div><div class="related panel pjax" data-title="系列文章"></div><div class="overview panel" data-title="站点概览"><div class="author" 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